**June 17, 2019 at 10:00 - June 21, 2019**
- BCAM

**Renato Lucà (Universität Basel)**

**DATE:** 17-21 June 2019 (5 sessions)

**TIME:** 10:00 - 12:00 (a total of 10 hours)

**LOCATION:** BCAM Seminar room

We will discuss the techniques used in [DGL17], [DZ18] to (almost) solve the problem of pointwise convergence of the Schrödinger equation to the initial data.

**PROGRAMME:**
Denoting with eit∆ f the solution of the linear Schrödinger equation on Rn with initial datum f, a very delicate problem is to identify the smallest regularity s ≥ 0 such that:

One can show by elementary methods that this convergence holds at any x ∈ Rn when f ∈ Hs and s > n/2, so the problem is interesting when s ≤ n/2. This low regularity regime has been first studied by Carleson [Car80], who proved the sufficiency of the condition s ≥ 1/4 in one dimension. Using a very simple counterexample, namely a superposition of wave packets, Dahlberg–Kenig [DK82] proved that for any s < 1/4 there are initial data f ∈ Hs for which the solution eit∆ f does not converge pointwise to f on a set of full Lebesgue measure. In particular, s ≥ 1/4 is necessary and sufficient for (1) in one dimension. The situation in higher dimensions is far more complicated. For instance, in dimensions n ≥ 3, no improvements to the sufficient condition s > 1/2, independently obtained by Vega [Veg88] and Sjo ̈lin [Sjo ̈87], have been obtained for a long time. However, recently Bourgain [Bou16] showed that s ≥ n/(2n + 2) is necessary and then s > n/(2n + 2) has been proved to be also sufficient in two dimensions by Du–Guth–Li [DGL17] and in higher dimensions by Du–Zhang [DZ18], so that the problem is solved (modulo endpoints).

We will give an introduction to the higher dimensional techniques introduced in [DGL17], [DZ18]. These techniques find applications also in related problems in Harmonic Analysis.

**REFERENCES: **
[Bou16] J. Bourgain. A note on the Schrödinger maximal function. J. Anal. Math., 130:393–396, 2016.

[Car80] L. Carleson. Some analytic problems related to statistical mechanics. In Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), volume 779 of Lecture Notes in Math., pages 5–45. Springer, Berlin, 1980.

[DGL17] X. Du, L. Guth, and X. Li. A sharp Schrödinger maximal estimate in R2. Ann. of Math. (2), 186(2):607–640, 2017.

[DK82] B. E. J. Dahlberg and C. E. Kenig. A note on the almost everywhere behavior of solutions to the Schrödinger equation. In Harmonic analysis (Minneapolis, Minn., 1981), volume 908 of Lecture Notes in Math., pages 205–209. Springer, Berlin-New York, 1982.

[DZ18] X. Du and R. Zhang. Sharp l2 estimate of Schrödinger maximal function in higher dimensions, 2018, arXiv:1805.02775.

[Sjo ̈87] P. Sjo ̈lin. Regularity of solutions to the Schrödinger equation. Duke Math. J., 55(3):699–715, 1987.

[Veg88] L. Vega. Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc., 102(4):874–878, 1988.

***Registration is free, but mandatory before June 14th: **To sign-up go to

https://bit.ly/2Gzyfeu and fill the registration form. Student grants are available. Please, let us know if you need support for travel and accommodation expenses when you fill the form.